This is done by first importing the existing web server data file into the new survey's DBF database. Then, when publishing the new project, you can choose the initialize web site with project data option available on the web form setup window.
The new project can share the name of the old project and should contain all of the field names from the previous project (unless you have made a decision to discard data fields).
Transitioning from the old iteration to the new iteration should be done at a time when the survey is not being accessed since data submitted during the transition will not be retained. Most transitions can be carried out in under 30 minutes in most cases, so the down time is minimal.
This is not a task to carry out without foresight, planning and a test run. You should always perform a backup before hand. Your steps should also be documented for future changes.